Chance-Constrained Combinatorial Optimization with a Probability Oracle and Its Application to Probabilistic Partial Set Covering

We investigate a class of chance-constrained combinatorial optimization problems. Given a pre-specified risk level \(\epsilon \in [0,1]\), the chance-constrained program aims to find the minimum cost selection of a vector of binary decisions \(x\) such that a desirable event \(\mathcal{B}(x)\) occur...

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Vydané v:arXiv.org
Hlavní autori: Hao-Hsiang, Wu, Kucukyavuz, Simge
Médium: Paper
Jazyk:English
Vydavateľské údaje: Ithaca Cornell University Library, arXiv.org 28.06.2018
Predmet:
Algorithms
Combinatorial analysis
Minimum cost
Optimization
Sampling
Statistical analysis
Substructures
ISSN:2331-8422
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Shrnutí:We investigate a class of chance-constrained combinatorial optimization problems. Given a pre-specified risk level \(\epsilon \in [0,1]\), the chance-constrained program aims to find the minimum cost selection of a vector of binary decisions \(x\) such that a desirable event \(\mathcal{B}(x)\) occurs with probability at least \( 1-\epsilon\). In this paper, we assume that we have an oracle that computes \(\mathbb P( \mathcal{B}(x))\) exactly. Using this oracle, we propose a general exact method for solving the chance-constrained problem. In addition, we show that if the chance-constrained program is solved approximately by a sampling-based approach, then the oracle can be used as a tool for checking and fixing the feasibility of the optimal solution given by this approach. We demonstrate the effectiveness of our proposed methods on a variant of the probabilistic set covering problem (PSC), which admits an efficient probability oracle. We give a compact mixed-integer program that solves PSC optimally (without sampling) for a special case. For large-scale instances for which the exact methods exhibit slow convergence, we propose a sampling-based approach that exploits the special structure of PSC. In particular, we introduce a new class of facet-defining inequalities for a submodular substructure of PSC, and show that a sampling-based algorithm coupled with the probability oracle solves the large-scale test instances effectively.
Bibliografia:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1708.02505